Answer: A student with a test grade of 42, the estimated homework grade is 42.
To find the linear regression equation, we need to calculate the slope (m) and y-intercept (b).
The slope represents the rate of change of the dependent variable (y) with respect to the independent variable (x), while the y-intercept represents the value of y when x is 0.
To calculate the slope (m), we use the formula: m = (Σ(xy) - (Σx)(Σy)/n) / (Σ(x^2) - ((Σx)^2)/n) where Σ represents the sum of the values, n is the number of data points, xy represents the product of x and y, and x^2 represents the square of x.
Let's calculate the values needed for the formula: Σx = 75 + 80 + 90 + 95 + 100 = 440 Σy = 82 + 85 + 92 + 88 + 95 = 442 Σxy = (75*82) + (80*85) + (90*92) + (95*88) + (100*95) = 41880 Σ(x^2) = (75^2) + (80^2) + (90^2) + (95^2) + (100^2) = 41650 n = 5
Substituting these values into the formula, we have: m = (41880 - (440*442)/5) / (41650 - (440^2)/5) m = (41880 - 97520/5) / (41650 - 96800/5) m = (41880 - 19504) / (41650 - 19360) m = 22376 / 22290 m ≈ 1
Now, let's calculate the y-intercept (b) using the formula: b = (Σy - m(Σx))/n
Substituting the values we have: b = (442 - 1(440))/5 b = (442 - 440)/5 b = 2/5 b ≈ 0.4
The linear regression equation is y = mx + b. Plugging in the values we calculated, the equation becomes: y = 1x + 0.4 y = x + 0.4
To estimate the homework grade for a student with a test grade of 42, we substitute the test grade (x) into the equation: y = 42 + 0.4 y ≈ 42.4 Rounding the estimated homework grade to the nearest integer, we get: Estimated homework grade ≈ 42 Therefore, for a student with a test grade of 42, the estimated homework grade is 42.